Problem 45 - Intermediate Algebra

Question

Add or subtract, and then simplify, if possible. See Example 4. $$\frac{n+2}{n-4}-\frac{n+5}{n+4}$$

Step-by-Step Solution

Answer

The simplified result is \(\frac{5n + 28}{(n-4)(n+4)}\).

Step 1: Identify the Operation

The operation we need to perform is subtraction of two rational expressions: \( \frac{n+2}{n-4} - \frac{n+5}{n+4} \).

Step 2: Find the Least Common Denominator (LCD)

The denominators of our fractions are \( n-4 \) and \( n+4 \). The LCD of these fractions is \((n-4)(n+4)\). We need this LCD to rewrite both fractions with a common denominator.

Step 3: Rewrite the Fractions with the Common Denominator

Rewrite each fraction with the common denominator \((n-4)(n+4)\). For \( \frac{n+2}{n-4} \), multiply both the numerator and the denominator by \( n+4 \), resulting in \( \frac{(n+2)(n+4)}{(n-4)(n+4)} \). For \( \frac{n+5}{n+4} \), multiply both the numerator and the denominator by \( n-4 \), giving \( \frac{(n+5)(n-4)}{(n+4)(n-4)} \).

Step 4: Perform the Subtraction

Subtract the fractions with the common denominator: \[ \frac{(n+2)(n+4)}{(n-4)(n+4)} - \frac{(n+5)(n-4)}{(n+4)(n-4)}. \] Do this by subtracting their numerators: \( (n+2)(n+4) - (n+5)(n-4) \).

Step 5: Apply the Distributive Property

Apply the distributive property to both products in the numerators. \((n+2)(n+4)\) expands to \(n^2 + 4n + 2n + 8\), which simplifies to \(n^2 + 6n + 8\). \((n+5)(n-4)\) expands to \(n^2 - 4n + 5n - 20\), which simplifies to \(n^2 + n - 20\).

Step 6: Combine Like Terms in the Numerator

Subtract the expanded expressions: \((n^2 + 6n + 8) - (n^2 + n - 20)\). This simplifies to: \[n^2 + 6n + 8 - n^2 - n + 20 = 5n + 28.\]

Step 7: Write the Final Simplified Expression

The simplified expression becomes \( \frac{5n + 28}{(n-4)(n+4)} \). This is the final result and cannot be simplified further.

Key Concepts

Least Common Denominator

When subtracting rational expressions, identifying and using the least common denominator, or LCD, is an essential step. The LCD is the smallest expression that is a multiple of each of the original denominators in your rational expressions. It allows you to rewrite the fractions so that they share a common denominator. This helps in performing addition or subtraction of those fractions easily.
To find the LCD of the two expressions \( \frac{n+2}{n-4} \) and \( \frac{n+5}{n+4} \), you first observe the denominators: \( n-4 \) and \( n+4 \). Since these are different, their product, \((n-4)(n+4)\), becomes their least common denominator. Be sure to always multiply different denominators together when they do not have common factors already. This step is crucial in ensuring the fractions can be combined accurately.

Distributive Property

The distributive property is a fundamental algebraic principle used to simplify expressions. It allows you to multiply a single term by each term inside a parenthesis, ensuring nothing is neglected.
In the expression \( (n+2)(n+4) \), when applying the distributive property, you compute it as follows: multiply \( n \) by \( n+4 \) and \( 2 \) by \( n+4 \). This results in:

• \( n \times n = n^2 \)
• \( n \times 4 = 4n \)
• \( 2 \times n = 2n \)
• \( 2 \times 4 = 8 \)

Combine the results to get \( n^2 + 4n + 2n + 8 \), which simplifies to \( n^2 + 6n + 8 \). You perform similar calculations for \( (n+5)(n-4) \), simplifying to \( n^2 + n - 20 \). Using the distributive property helps break down complex expressions into manageable parts, making the subtraction straightforward.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a key skill when working with rational expressions. After finding a common denominator and using the distributive property, you'll often need to combine like terms and consolidate the expression.
For example, in our problem, once you have the expanded forms of the numerators: \( n^2 + 6n + 8 \) and \( n^2 + n - 20 \), you need to subtract them. Take:

• \( n^2 + 6n + 8 - n^2 - n + 20 \)

Notice how the \( n^2 \) terms cancel each other out, simplifying the subtraction to just the linear and constant terms. This becomes \( 5n + 28 \). Thus, the final simplified expression is \( \frac{5n + 28}{(n-4)(n+4)} \). Simplification makes it easier to understand the expression and see whether it can be simplified more. It provides a clean and clear final result, essential for interpreting and using algebraic expressions.